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A001897 Denominators of cosecant numbers: -2*(2^(2*n-1)-1)*Bernoulli(2*n).

%I A001897 M2983 N1205 #108 Jan 24 2024 09:16:14
%S A001897 1,3,15,21,15,33,1365,3,255,399,165,69,1365,3,435,7161,255,3,959595,3,
%T A001897 6765,903,345,141,23205,33,795,399,435,177,28393365,3,255,32361,15,
%U A001897 2343,70050435,3,15,1659,115005,249,1702155,3,30705,136059,705,3,2250885,3,16665,2163
%N A001897 Denominators of cosecant numbers: -2*(2^(2*n-1)-1)*Bernoulli(2*n).
%C A001897 Same as half the denominators of the even-indexed Bernoulli numbers B_{2*n} for n>0, by the von Staudt-Clausen theorem and Fermat's little theorem. - _Bernd C. Kellner_ and _Jonathan Sondow_, Jan 02 2017 [This is implemented in the second Maple program. - _Peter Luschny_, Aug 21 2021]
%D A001897 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 187.
%D A001897 S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51.
%D A001897 N. E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 458.
%D A001897 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.
%D A001897 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001897 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001897 Hector Blandin and Rafael Diaz, <a href="http://arXiv.org/abs/0708.0809">Compositional Bernoulli numbers</a>, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897.
%H A001897 S. A. Joffe, <a href="/A001896/a001896.pdf">Sums of like powers of natural numbers</a>, Quart. J. Pure Appl. Math. 46 (1914), 33-51. [Annotated scanned copy of pages 38-51 only, plus notes]
%H A001897 Masanobu Kaneko, Maneka Pallewatta, and Hirofumi Tsumura, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Tsumura/tsumura3.html">On Polycosecant Numbers</a>, J. Integer Seq. 23 (2020), no. 6, 17 pp. See line k=1 of Table 1 p. 3.
%H A001897 D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.
%H A001897 N. E. Nörlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen über Differenzenrechnung</a>, Springer 1924, p. 27.
%H A001897 N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
%F A001897 a(0)=1, a(n)=(1/2)*A002445(n) for n>=1. - _Joerg Arndt_, May 07 2012
%F A001897 a(n) = denominator((2*n)!*Li_{2*n}(1)) for n > 0. - _Peter Luschny_, Jun 29 2012
%F A001897 a(0)=1, a(n) = (1/2)*A027642(2*n) = (3/2)*A277087(n) for n>=1. - _Jonathan Sondow_, Dec 14 2016
%F A001897 From _Peter Luschny_, Sep 06 2017: (Start)
%F A001897 a(n) = denominator(r(n)) where r(n) = Sum_{0..n} (-1)^(n-k)*A241171(n, k)/(2*k+1).
%F A001897 a(n) = denominator(bernoulli(2*n, 1/2))/4^n = A033469(n)/4^n. (End)
%F A001897 Apparently a(n) = denominator(Sum_{k=0..2*n-2} (-1)^k*E2(2*n-1, k+1)/binomial(4*n-1, k+1)), where E2(n, k) denotes the second-order Eulerian numbers A340556. - _Peter Luschny_, Feb 17 2021
%e A001897 Cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)} are 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255, -5749691557/399, 91546277357/165, -1792042792463/69, 1982765468311237/1365, -286994504449393/3, 3187598676787461083/435, ... = A001896/A001897.
%p A001897 b := n -> bernoulli(n)*2^add(i,i=convert(n,base,2));
%p A001897 a := n -> denom(b(2*n)); # _Peter Luschny_, May 02 2009
%p A001897 # Alternative :
%p A001897 Clausen := proc(n) local i,S; map(i->i+1, numtheory[divisors](n));
%p A001897 S := select(isprime, %); if S <> {} then mul(i,i=S) else NULL fi end:
%p A001897 A001897_list := n -> [1,seq(Clausen(2*i)/2,i=1..n-1)];
%p A001897 A001897_list(52); # _Peter Luschny_, Oct 03 2011
%t A001897 a[n_] := Denominator[-2*(2^(2*n-1)-1)*BernoulliB[2*n]]; Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, Sep 11 2013 *)
%o A001897 (Sage)
%o A001897 def A001897(n):
%o A001897     if n == 0:
%o A001897         return 1
%o A001897     M = (d + 1 for d in divisors(2 * n))
%o A001897     return prod(s for s in M if is_prime(s)) / 2
%o A001897 [A001897(n) for n in range(55)]  # _Peter Luschny_, Feb 20 2016
%o A001897 (PARI) a(n) = denominator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ _Michel Marcus_, Apr 06 2019
%o A001897 (Magma) [Denominator(2*(1-2^(2*n-1))*Bernoulli(2*n)): n in [0..55]]; // _G. C. Greubel_, Apr 06 2019
%Y A001897 Cf. A001896 (numerators), A027642, A033469, A160014, A241171, A277087, A340556.
%Y A001897 Cf. A132092-A132099,
%K A001897 nonn,frac
%O A001897 0,2
%A A001897 _N. J. A. Sloane_